Subgroups — C₂²×C₇×#462 #13237623

Order 1792.

2790 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₁₆	C₂	1	2	yes	maximal
2	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#462	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×#103	C₂	1	2	yes	maximal
7	C₂³×C₇×Q₈	C₂²	1	2	yes	—
8	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇×Q₁₆	—	2	16	no	—
10	C₂²×C₇×Q₁₆	C₂²	1	24	yes	—
11	C₂³×C₇×C₈	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₇×C₄⋊Q₈	C₂²	1	12	yes	—
14	C₂×C₄×C₇×Q₈	—	2	6	no	—
15	C₂×C₄×C₇×Q₈	C₂²	1	12	yes	—
16	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
17	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
18	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
19	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₇	—	2	1	no	—
21	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#462	C₂²	1	64	yes	direct product factor
23	C₂×C₇×C₄⋊C₈	C₂²	1	12	yes	—
24	C₂×C₇×#103	C₂²	1	24	yes	—
25	C₂²×#462	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₇×Q₈	—	2	3	no	—
27	C₂²×C₇×Q₈	—	2	6	no	—
28	C₂²×C₇×Q₈	—	2	4	no	—
29	C₂²×C₇×Q₈	—	2	8	no	—
30	C₂²×C₇×Q₈	—	2	8	no	—
31	C₂²×C₇×Q₈	D₄	1	8	yes	—
32	C₂²×C₇×Q₈	C₂³	1	12	yes	—
33	C₂²×C₇×Q₈	C₂³	1	6	yes	—
34	C₂³×C₄×C₇	—	2	1	no	—
35	C₂³×C₄×C₇	—	2	2	no	—
36	C₂³×C₄×C₇	—	2	1	no	—
37	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₇×Q₁₆	—	2	96	no	—
41	C₂×C₇×Q₁₆	C₂³	1	32	yes	—
42	C₂²×C₇×C₈	—	2	8	no	—
43	C₂²×C₇×C₈	C₂³	1	12	yes	—
44	C₇×C₄⋊Q₈	—	2	24	no	—
45	C₇×C₄⋊Q₈	C₂³	1	16	yes	—
46	C₄×C₇×Q₈	—	2	24	no	—
47	C₄×C₇×Q₈	C₂³	1	16	yes	—
48	C₂×C₇×C₄⋊C₄	—	2	12	no	—
49	C₂×C₇×C₄⋊C₄	—	2	12	no	—
50	C₂×C₇×C₄⋊C₄	—	2	12	no	—
51	C₂×C₇×C₄⋊C₄	—	2	3	no	—
52	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
53	C₂×C₄²×C₇	—	2	12	no	—
54	C₂×C₄²×C₇	D₄	1	6	yes	—
55	C₂×C₄²×C₇	C₂³	1	6	yes	—
56	C₇×C₄⋊C₈	C₂³	1	16	yes	—
57	C₇×#103	C₂³	1	32	yes	—
58	C₂³×Q₁₆	C₂×C₇	1	2	yes	—
59	C₂²×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
60	C₂²×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
61	C₂×#462	C₂×C₇	1	24	yes	direct product factor
62	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
63	C₂²×#103	C₂×C₇	1	2	yes	—
64	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×Q₈	—	2	24	no	—
66	C₂×C₇×Q₈	—	2	6	no	—
67	C₂×C₇×Q₈	—	2	12	no	—
68	C₂×C₇×Q₈	—	2	12	no	—
69	C₂×C₇×Q₈	—	2	24	no	—
70	C₂×C₇×Q₈	—	2	24	no	—
71	C₂×C₇×Q₈	—	4	24	no	—
72	C₂×C₇×Q₈	C₂×D₄	1	24	yes	—
73	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
74	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₇	—	2	8	no	—
76	C₂²×C₄×C₇	—	2	4	no	—
77	C₂²×C₄×C₇	—	2	6	no	—
78	C₂²×C₄×C₇	—	2	6	no	—
79	C₂²×C₄×C₇	—	2	12	no	—
80	C₂²×C₄×C₇	—	4	8	no	—
81	C₂²×C₄×C₇	—	4	4	no	—
82	C₂²×C₄×C₇	Q₁₆	1	8	yes	complemented
83	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
87	C₂²×C₄×C₇	#41	1	4	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
89	C₇×Q₁₆	—	2	64	no	—
90	C₂×C₇×C₈	—	2	24	no	—
91	C₂×C₇×C₈	C₂⁴	1	8	yes	—
92	C₇×C₄⋊C₄	—	2	16	no	—
93	C₇×C₄⋊C₄	—	2	16	no	—
94	C₇×C₄⋊C₄	—	2	16	no	—
95	C₇×C₄⋊C₄	—	2	6	no	—
96	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
97	C₄²×C₇	—	2	16	no	—
98	C₄²×C₇	C₂×D₄	1	12	yes	—
99	C₄²×C₇	C₂⁴	1	4	yes	—
100	C₂³×Q₈	C₂²×C₇	1	2	yes	—
101	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
102	C₂²×Q₁₆	—	2	16	no	—
103	C₂²×Q₁₆	C₂²×C₇	1	24	yes	—
104	C₂³×C₈	C₂²×C₇	1	2	yes	—
105	C₂×C₄⋊Q₈	—	2	6	no	—
106	C₂×C₄⋊Q₈	C₂²×C₇	1	12	yes	—
107	C₂×C₄×Q₈	—	2	6	no	—
108	C₂×C₄×Q₈	C₂²×C₇	1	12	yes	—
109	C₂²×C₄⋊C₄	—	2	1	no	—
110	C₂²×C₄⋊C₄	—	2	1	no	—
111	C₂²×C₄⋊C₄	—	2	1	no	—
112	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
113	C₂²×C₄²	—	2	1	no	—
114	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
115	#462	C₂²×C₇	1	64	yes	direct product factor
116	C₂×C₄⋊C₈	C₂²×C₇	1	12	yes	—
117	C₂×#103	C₂²×C₇	1	24	yes	—
118	C₂³×C₇	#121	1	4	yes	—
119	C₂³×C₇	C₂×Q₁₆	1	4	yes	—
120	C₂³×C₇	#137	1	4	yes	—
121	C₂³×C₇	C₂²×D₄	1	3	yes	—
122	C₇×Q₈	—	2	24	no	—
123	C₇×Q₈	—	2	4	no	—
124	C₇×Q₈	—	2	8	no	—
125	C₇×Q₈	—	2	8	no	—
126	C₇×Q₈	—	4	24	no	—
127	C₇×Q₈	C₂²×D₄	1	8	yes	—
128	C₂×C₄×C₇	—	2	24	no	—
129	C₂×C₄×C₇	—	2	12	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	2	8	no	—
133	C₂×C₄×C₇	—	4	24	no	—
134	C₂×C₄×C₇	—	4	12	no	—
135	C₂×C₄×C₇	#121	1	12	yes	—
136	C₂×C₄×C₇	C₂×Q₁₆	1	24	yes	complemented
137	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
138	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
139	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
140	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
141	C₂×C₄×C₇	C₂×#41	1	6	yes	—
142	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
143	C₇×C₈	—	2	8	no	—
144	C₂²×Q₈	—	2	3	no	—
145	C₂²×Q₈	—	2	6	no	—
146	C₂²×Q₈	—	2	4	no	—
147	C₂²×Q₈	—	2	8	no	—
148	C₂²×Q₈	—	2	8	no	—
149	C₂²×Q₈	C₇×D₄	1	8	yes	—
150	C₂²×Q₈	C₂³×C₇	1	12	yes	—
151	C₂²×Q₈	C₂³×C₇	1	6	yes	—
152	C₂³×C₄	—	2	1	no	—
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
156	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
157	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	96	no	—
159	C₂×Q₁₆	C₂³×C₇	1	32	yes	—
160	C₂²×C₈	—	2	8	no	—
161	C₂²×C₈	C₂³×C₇	1	12	yes	—
162	C₄⋊Q₈	—	2	24	no	—
163	C₄⋊Q₈	C₂³×C₇	1	16	yes	—
164	C₄×Q₈	—	2	24	no	—
165	C₄×Q₈	C₂³×C₇	1	16	yes	—
166	C₂×C₄⋊C₄	—	2	12	no	—
167	C₂×C₄⋊C₄	—	2	12	no	—
168	C₂×C₄⋊C₄	—	2	12	no	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
171	C₂×C₄²	—	2	12	no	—
172	C₂×C₄²	C₇×D₄	1	6	yes	—
173	C₂×C₄²	C₂³×C₇	1	6	yes	—
174	C₄⋊C₈	C₂³×C₇	1	16	yes	—
175	#103	C₂³×C₇	1	32	yes	—
176	C₂²×C₇	#462	1	16	yes	direct product factor
177	C₂²×C₇	C₂×#121	1	6	yes	—
178	C₂²×C₇	C₂²×Q₁₆	1	6	yes	—
179	C₂²×C₇	C₂×#137	1	6	yes	—
180	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
181	C₄×C₇	—	2	8	no	—
182	C₄×C₇	—	2	4	no	—
183	C₄×C₇	—	4	8	no	—
184	C₄×C₇	—	4	4	no	—
185	C₄×C₇	C₂×#121	1	6	yes	—
186	C₄×C₇	C₂²×Q₁₆	1	8	yes	complemented
187	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
188	C₄×C₇	C₂²×#41	1	1	yes	characteristic
189	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
190	C₂×Q₈	—	2	24	no	—
191	C₂×Q₈	—	2	6	no	—
192	C₂×Q₈	—	2	12	no	—
193	C₂×Q₈	—	2	12	no	—
194	C₂×Q₈	—	2	24	no	—
195	C₂×Q₈	—	2	24	no	—
196	C₂×Q₈	—	4	24	no	—
197	C₂×Q₈	C₂×C₇×D₄	1	24	yes	—
198	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
199	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	6	no	—
203	C₂²×C₄	—	2	6	no	—
204	C₂²×C₄	—	2	12	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₇×Q₁₆	1	8	yes	complemented
208	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
209	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
210	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
211	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
212	C₂²×C₄	C₇×#41	1	4	yes	—
213	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
214	Q₁₆	—	2	64	no	—
215	C₂×C₈	—	2	24	no	—
216	C₂×C₈	C₂⁴×C₇	1	8	yes	—
217	C₄⋊C₄	—	2	16	no	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	6	no	—
221	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
222	C₄²	—	2	16	no	—
223	C₄²	C₂×C₇×D₄	1	12	yes	—
224	C₄²	C₂⁴×C₇	1	4	yes	—
225	C₂×C₇	C₂×#462	1	12	yes	direct product factor
226	C₂×C₇	C₂²×#121	1	1	yes	characteristic
227	C₂×C₇	C₂³×Q₁₆	1	1	yes	characteristic
228	C₂×C₇	C₂²×#137	1	1	yes	characteristic
229	C₂³	C₇×#121	1	4	yes	—
230	C₂³	C₂×C₇×Q₁₆	1	4	yes	—
231	C₂³	C₇×#137	1	4	yes	—
232	C₂³	C₂²×C₇×D₄	1	3	yes	—
233	Q₈	—	2	4	no	—
234	Q₈	—	2	24	no	—
235	Q₈	—	2	8	no	—
236	Q₈	—	2	8	no	—
237	Q₈	—	4	24	no	—
238	Q₈	C₂²×C₇×D₄	1	8	yes	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	24	no	—
241	C₂×C₄	—	2	12	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	24	no	—
245	C₂×C₄	—	4	12	no	—
246	C₂×C₄	C₇×#121	1	12	yes	—
247	C₂×C₄	C₂×C₇×Q₁₆	1	24	yes	complemented
248	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
249	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
250	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
251	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
252	C₂×C₄	C₂×C₇×#41	1	6	yes	—
253	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
254	C₈	—	2	8	no	—
255	C₇	C₂²×#462	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²	C₇×#462	1	16	yes	direct product factor
257	C₂²	C₂×C₇×#121	1	6	yes	—
258	C₂²	C₂²×C₇×Q₁₆	1	6	yes	—
259	C₂²	C₂×C₇×#137	1	6	yes	—
260	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	—	4	4	no	—
264	C₄	—	4	8	no	—
265	C₄	C₂×C₇×#121	1	6	yes	—
266	C₄	C₂²×C₇×Q₁₆	1	8	yes	complemented
267	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
268	C₄	C₂²×C₇×#41	1	1	yes	characteristic
269	C₂	C₂×C₇×#462	1	12	yes	direct product factor
270	C₂	C₂²×C₇×#121	1	1	yes	characteristic
271	C₂	C₂³×C₇×Q₁₆	1	1	yes	characteristic
272	C₂	C₂²×C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₂×C₇×#462	C₂	direct (N × H)
22. C₇×#462	C₂²	direct (N × H)
25. C₂²×#462	C₇	direct (N × H)
61. C₂×#462	C₂×C₇	direct (N × H)
82. C₂²×C₄×C₇	Q₁₆	semidirect
115. #462	C₂²×C₇	direct (N × H)
136. C₂×C₄×C₇	C₂×Q₁₆	semidirect
176. C₂²×C₇	#462	direct (N × H)
186. C₄×C₇	C₂²×Q₁₆	semidirect
207. C₂²×C₄	C₇×Q₁₆	semidirect
225. C₂×C₇	C₂×#462	direct (N × H)
247. C₂×C₄	C₂×C₇×Q₁₆	semidirect
255. C₇	C₂²×#462	direct (N × H)
256. C₂²	C₇×#462	direct (N × H)
266. C₄	C₂²×C₇×Q₁₆	semidirect
269. C₂	C₂×C₇×#462	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₇×Q₁₆	2
C₂²×C₇×C₄⋊Q₈	2
C₂²×C₄×C₇×Q₈	2
C₂×C₇×#462	2
C₂²×C₇×C₄⋊C₈	2
C₂²×C₇×#103	2
C₂²×#462	7

Subgroups — C22×C7×#462 #13237623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#462 #13237623